Unformatted text preview: of the edge S , where · is a matrix norm, and 1 is a vector of ones of length |S |. (a) Show that the optimization problem
minimize edges S fS ( X ) is convex in the free node coordinates xj .
(b) The size fS (X ) of a net S obviously depends on the norm used in the deﬁnition (41). We
consider ﬁve norms.
• Frobenius norm:
Xs − y 1T F = j ∈S i=1 • Maximum Euclidean column norm: 1/2 p (xij − yi )2 1/2 p X S − y 1T = max 2,1 j ∈S i=1 • Maximum column sum norm: (xij − yi )2 p X S − y 1T 1,1 = max
j ∈S i=1 |xij − yi |. • Sum of absolute values norm:
p X s − y 1T sav 127 . =
j ∈S i=1 |xij − yi | . • Sum-row-max norm: p X s − y 1T srm =
i=1 max |xij − yi |
j ∈S For which of these norms does fS have the following interpretations?
(i) fS (X ) is the radius of the smallest Euclidean ball that contains the nodes of S .
(ii) fS (X ) is (proportional to) the perimeter of the smallest rectangle that contains the nodes
of S :
fS ( X ) =
(max xij − min xij ).
4 i=1 j ∈S
(iii) fS (X ) is the squ...
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This note was uploaded on 09/10/2013 for the course C 231 taught by Professor F.borrelli during the Fall '13 term at Berkeley.
- Fall '13
- The Aeneid